Chapter 16. Additional Regression Methods

462—Chapter 16. Additional Regression Methods

The PDL is sometimes referred to as an Almon lag. The constant c is included only to avoid numerical problems that can arise from collinearity and does not affect the estimates of β .

This specification allows you to estimate a model with k lags of x using only p parameters (if you choose p > k , EViews will return a “Near Singular Matrix” error).

(16.5)

…

zp + 1 = ( −c)pxt + ( 1 − c)pxt − 1 + … + ( k − c)pxt − k

Once we estimate γ from Equation (16.4), we can recover the parameters of interest β , and their standard errors using the relationship described in Equation (16.2). This procedure is straightforward since β is a linear transformation of γ .

The specification of a polynomial distributed lag has three elements: the length of the lag k , the degree of the polynomial (the highest power in the polynomial) p , and the constraints that you want to apply. A near end constraint restricts the one-period lead effect of x on y to be zero:

A far end constraint restricts the effect of x on y to die off beyond the number of specified lags:

If you restrict either the near or far end of the lag, the number of γ parameters estimated is reduced by one to account for the restriction; if you restrict both the near and far end of the lag, the number of γ parameters is reduced by two.

By default, EViews does not impose constraints.

How to Estimate Models Containing PDLs

You specify a polynomial distributed lag by the pdl term, with the following information in parentheses, each separated by a comma in this order:

•The name of the series.

•The lag length (the number of lagged values of the series to be included).

Special Equation Terms—463

•The degree of the polynomial.

•A numerical code to constrain the lag polynomial (optional):

You may omit the constraint code if you do not want to constrain the lag polynomial. Any number of pdl terms may be included in an equation. Each one tells EViews to fit distributed lag coefficients to the series and to constrain the coefficients to lie on a polynomial.

For example, the commands:

ls sales c pdl(orders,8,3)

fits SALES to a constant, and a distributed lag of current and eight lags of ORDERS, where the lag coefficients of ORDERS lie on a third degree polynomial with no endpoint constraints. Similarly:

ls div c pdl(rev,12,4,2)

fits DIV to a distributed lag of current and 12 lags of REV, where the coefficients of REV lie on a 4th degree polynomial with a constraint at the far end.

The pdl specification may also be used in two-stage least squares. If the series in the pdl is exogenous, you should include the PDL of the series in the instruments as well. For this purpose, you may specify pdl(*) as an instrument; all pdl variables will be used as instruments. For example, if you specify the TSLS equation as,

sales c inc pdl(orders(-1),12,4)

with instruments:

fed fed(-1) pdl(*)

the distributed lag of ORDERS will be used as instruments together with FED and FED(–1).

Polynomial distributed lags cannot be used in nonlinear specifications.

Example

The distributed lag model of industrial production (IP) on money (M1) yields the following results:

464—Chapter 16. Additional Regression Methods

Dependent Variable: IP

Method: Least Squares

Date: 08/15/97 Time: 17:09

Sample(adjusted): 1960:01 1989:12

Included observations: 360 after adjusting endpoints

Taken individually, none of the coefficients on lagged M1 are statistically different from zero. Yet the regression as a whole has a reasonable R2 with a very significant F-statistic (though with a very low Durbin-Watson statistic). This is a typical symptom of high collinearity among the regressors and suggests fitting a polynomial distributed lag model.

To estimate a fifth-degree polynomial distributed lag model with no constraints, set the sample using the command,

smpl 1959:01 1989:12

then estimate the equation specification:

ip c pdl(m1,12,5)

using a command with the specification, or by entering the specification in the Equation Estimation dialog.

The following result is reported at the top of the equation window:

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the joint significance of these two terms, click View/Coefficient Tests/Wald-Coefficient Restrictions… and enter:

c(6)=0, c(7)=0

in the Wald Test dialog box (see “Wald Test (Coefficient Restrictions)” on page 572 for an extensive discussion of Wald tests in EViews). EViews displays the result of the joint test:

Wald Test:

Equation: IP_PDL

Restrictions are linear in coefficients.

There is no evidence to reject the null hypothesis, suggesting that you could have fit a lower order polynomial to your lag structure.

Automatic Categorical Dummy Variables

EViews equation specifications support expressions of the form:

@EXPAND(ser1[, ser2, ser3, ...][, drop_spec])

that, when used in an equation specification, creates a set of dummy variables that span the unique integer or string values of the input series.

For example consider the following two variables:

•SEX is a numeric series which takes the values 1 and 0.

•REGION is an alpha series which takes the values “North”, “South”, “East”, and “West”.

The equation list specification

income age @expand(sex)

is used to regress INCOME on the regressor AGE, and two dummy variables, one for “SEX=0” and one for “SEX=1”.

Similarly, the @EXPAND statement in the equation list specification,

468—Chapter 16. Additional Regression Methods

We caution you to take some care in using @EXPAND since it is very easy to generate excessively large numbers of regressors.

Example

Following Wooldridge (2000, Example 3.9, p. 106), we regress the log median housing price, LPRICE, on a constant, the log of the amount of pollution (LNOX), and the average number of houses in the community, ROOMS, using data from Harrison and Rubinfeld (1978).

We expand the example to include a dummy variable for each value of the series RADIAL, representing an index for community access to highways. We use @EXPAND to create the dummy variables of interest, with a list specification of:

lprice lnox rooms @expand(radial)

We deliberately omit the constant term C since the @EXPAND creates a full set of dummy variables. The top portion of the results is depicted below:

Dependent Variable: LPRICE

Method: Least Squares

Date: 12/30/03 Time: 16:49

Sample: 1 506

Included observations: 506

Note that EViews has automatically created dummy variable expressions for each distinct value in RADIAL. If we wish to renormalize our dummy variables with respect to a different omitted category, we may include the C in the regression list, and explicitly exclude a value. For example, to exclude the category RADIAL=24, we use the list:

lprice c lnox rooms @expand(radial, @drop(24))

Estimation of this specification yields: